Another thing I would like to talk about is the following property.

I will start with an example.

Let x be a real.

Consider . If n goes to oo the limit gives .

Now intuitively one would expect about the derivative with respect to x that lim n-> oo

Clearly

However so lim n-> oo =/= 0 !

This is an important and classic lesson in math.

So many of our iterations used here require formal and carefull analysis.

If you are not convinced notice F(n,x) = g^[n](Q(n,x)) IS something that occurs in the majority of limits related to tetration including e.g. tommysexp , basechange , interpolation methods , matrix methods , iterations to improve on riemann mappings , ... !

I do not know if has been proven here for any method ?

Also note that I took x to be a real. So this is about Coo functions defined over the reals but for analytic functions defined over the complex number the issue is even bigger.

Also note that in my example both f(n,x) and f(x) were Coo , in fact even analytic. So being analytic or Coo does not solve this issue.

I think we need to work on this.

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For those clueness on how to such things , I point out that putting boundaries on values or signs and using the intermediate value theorem frequently does miracles.

I will start with an example.

Let x be a real.

Consider . If n goes to oo the limit gives .

Now intuitively one would expect about the derivative with respect to x that lim n-> oo

Clearly

However so lim n-> oo =/= 0 !

This is an important and classic lesson in math.

So many of our iterations used here require formal and carefull analysis.

If you are not convinced notice F(n,x) = g^[n](Q(n,x)) IS something that occurs in the majority of limits related to tetration including e.g. tommysexp , basechange , interpolation methods , matrix methods , iterations to improve on riemann mappings , ... !

I do not know if has been proven here for any method ?

Also note that I took x to be a real. So this is about Coo functions defined over the reals but for analytic functions defined over the complex number the issue is even bigger.

Also note that in my example both f(n,x) and f(x) were Coo , in fact even analytic. So being analytic or Coo does not solve this issue.

I think we need to work on this.

---

For those clueness on how to such things , I point out that putting boundaries on values or signs and using the intermediate value theorem frequently does miracles.